Heilbronn's conjecture on Waring's number (mod p)

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Let p be a prime k | p − 1 , t = ( p − 1 ) / k and γ ( k , p ) be the minimal value of s such that every number is a sum of s kth powers ( mod p ) . We prove Heilbronn's conjecture that γ ( k , p ) ≪ k 1 / 2 for t > 2 . More generally we show that for any positive integer q, γ ( k , p ) ⩽ C ( q ) k 1 / q for ϕ ( t ) ⩾ q . A comparable lower bound is also given. We also establish exact values for γ ( k , p ) when ϕ ( t ) = 2 . For instance, when t = 3 , γ ( k , p ) = a + b − 1 where a > b > 0 are the unique integers with a 2 + b 2 + a b = p , and when t = 4 , γ ( k , p ) = a − 1 where a > b > 0 are the unique integers with a 2 + b 2 = p .